President Obama and congressional Republicans say they are hopeful about new White House outreach efforts, but don't yet know whether they will translate to legislative reality.

After a flurry of phone calls, a private dinner with Republican senators and a White House lunch with a key House member, President Obama and the GOP remain divided over whether a new debt reduction plan should include new taxes.

"That's the challenge," said Sen. John Hoeven, R-N.D., one of a dozen Republican senators who dined with Obama on Wednesday. "That's exactly why we've got to stay at it."

Obama -- looking for an alternative to sequestration, $85 billion in automatic budget cuts that took effect March 1 -- lunched Thursday with Rep. Paul Ryan, R-Wis., chairman of the House Budget Committee and the GOP's 2012 vice presidential nominee. The committee's top Democrat, Rep. Chris Van Hollen of Maryland, also attended.

The lunch came one day after the presidential dinner with a dozen Republican senators and within days of Obama's phone calls to other GOP lawmakers.

Obama continues to push for a new debt reduction agreement that includes more targeted budget cuts and increased tax revenues by eliminating loopholes that benefit the wealthy; Republicans oppose higher taxes as part of any agreement, saying they would slow economic growth.

House Speaker John Boehner, R-Ohio, described the dinner and other outreach efforts as hopeful signs -- "but if the president continues to insist on tax hikes, I don't think we're going to get very far."

White House spokesman Jay Carney, meanwhile, said Obama will continue talking to Republicans about a "balanced" plan that includes both cuts and new revenues, so that middle class programs don't bear all the brunt of debt reduction.

"We'll see where these conversations lead," Carney said. "What the president hopes is that there is a spirit of compromise."

Carney said Obama is also talking with Republicans about other aspects of his agenda, including an immigration bill and legislation to battle gun violence.

For his part, Ryan thanked the president for inviting him to lunch and said in a statement that "everyone needs to be a part of this conversation. We need an open debate about how best to balance the budget and expand opportunity."

The total federal debt now exceeds $16.6 trillion.

In a telephone interview, Hoeven said the parties have a "four- to five-month window" to reach bipartisan agreement on budget issues. After that, they are likely to be sidetracked by discussions on raising the debt ceiling and early campaigning for the 2014 congressional elections.

The president needs to stay engaged and avoid public attacks on Republicans, Hoeven said. And Republicans need to make the case that debt reduction should focus on spending cuts.

The tax and spending dispute hung over Wednesday night's dinner with Obama and Republican senators at The Jefferson hotel in downtown Washington (and, yes, the White House confirmed that the president picked up the check personally, but did not provide an amount).

The menu featured a selection of appetizers that included golden beet soup and Maryland blue crab risotto, with entrees that ranged from Colorado lamb to lobster thermidor, according to the hotel. The dessert menu featured peanut butter crumble and "Heart of Guana Chocolate Tart."

Carney said Obama "found the dinner very constructive and very pleasant." Carney declined to provide details of the talks, saying the White House is looking "to help foster an environment where these conversations are productive."

Sen. John McCain, R-Ariz., the 2008 Republican presidential nominee, gave reporters a thumbs-up after the dinner at The Jefferson hotel.

McCain sat next to the president during the proceedings, participants said.

Another Republican diner, Sen. Pat Toomey of Pennsylvania, told Fox and Friends on Thursday, "I'm hoping this is a new approach on the part of the president."

The parties didn't strike a deal, but "that isn't going to happen over one dinner," Toomey said. "But I think it was a constructive exchange."

Other Republican senators who dined with Obama: Lindsey Graham of South Carolina, Kelly Ayotte of New Hampshire, Bob Corker of Tennessee, Dan Coats of Indiana, Tom Coburn of Oklahoma, Richard Burr of North Carolina, Mike Johanns of Nebraska, Ron Johnson of Wisconsin and Saxby Chambliss of Georgia.